| L(s) = 1 | − 1.54·2-s − 2.24·3-s + 0.375·4-s + 3.45·6-s − 3.95·7-s + 2.50·8-s + 2.02·9-s + 1.30·11-s − 0.841·12-s − 4.16·13-s + 6.10·14-s − 4.60·16-s + 5.17·17-s − 3.12·18-s + 5.17·19-s + 8.87·21-s − 2.00·22-s − 2.34·23-s − 5.61·24-s + 6.41·26-s + 2.17·27-s − 1.48·28-s + 5.27·29-s + 0.379·31-s + 2.09·32-s − 2.92·33-s − 7.98·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 1.29·3-s + 0.187·4-s + 1.41·6-s − 1.49·7-s + 0.885·8-s + 0.676·9-s + 0.393·11-s − 0.242·12-s − 1.15·13-s + 1.63·14-s − 1.15·16-s + 1.25·17-s − 0.737·18-s + 1.18·19-s + 1.93·21-s − 0.428·22-s − 0.488·23-s − 1.14·24-s + 1.25·26-s + 0.419·27-s − 0.280·28-s + 0.980·29-s + 0.0681·31-s + 0.370·32-s − 0.509·33-s − 1.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3164615710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3164615710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 0.379T + 31T^{2} \) |
| 37 | \( 1 - 0.812T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 + 0.246T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 6.57T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 - 9.26T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 4.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.042485635349315476216524542983, −7.26704750024574211005347057375, −6.81800898930827164990155211386, −6.02515793280754290902883128216, −5.35425485182120952632448758945, −4.65048625870019775679796829552, −3.60088930486647226951038616986, −2.69719005042910304959623945889, −1.24240666109510184382512341768, −0.42916285648824567578799943583,
0.42916285648824567578799943583, 1.24240666109510184382512341768, 2.69719005042910304959623945889, 3.60088930486647226951038616986, 4.65048625870019775679796829552, 5.35425485182120952632448758945, 6.02515793280754290902883128216, 6.81800898930827164990155211386, 7.26704750024574211005347057375, 8.042485635349315476216524542983
